Tauberian operators in \(p\)-adic analysis (Q1587324)

Let \(K\) be a non-Archimedean nontrivially valued complete field with a valuation \(|\cdot|\), \(E\) and \(F\) be infinite-dimensional Banach spaces over \(K\) and \(L(E,F)\) be the set of all continuous linear operators from \(E\) into \(F\). We say that \(T\in L(E,F)\) is semi-Fredholm if its range space \(R(T)\) is closed in \(F\) and its kernel \(N(T)\) is finite-dimensional. \(T\in L(E, F)\) is called Tauberian if \(x''\in E''\), \(T''x''\in E\) imply \(x''\in E\). \(T\in L(E, F)\) is said to have property N if \(x''\in E''\), \(T'' x''= 0\) imply \(x''\in E\). A sequence \(\{x_n\}_{n\geq 1}\) in \(E\) is said to be a basic sequence if \(\{x_n\}_{n\geq 1}\) is a basis for its closed linear \(\text{span}[\{x_i: i=1,2,\dots, n,\dots\}]\). In Archimedean analysis Tauberian operators having property N were defined by \textit{N. Kalton} and \textit{A. Wilansky} [Proc. Am. Math. Soc. 57, 251-255 (1976; Zbl 0304.47023)]. In this paper several characterization of \(p\)-adic Tauberian operators and operators having property N in terms of basic sequences are given. As its applications, some equivalent relations between these operators and \(p\)-adic semi-Fredholm operators are also given in the paper.